Algebraic structure and Poisson's theory of mechanico-electrical systems

Abstract

This letter focuses on studying algebraic structure and the Poisson's theory of mechanico-electrical systems. Based on the relationship between momentum and Hamiltonian, we present Hamilton canonical equations and generalized Hamilton canonical equations and their contravariant algebraic forms for mechanico-electrical systems. The work derives that Lagrange mechanico-electrical systems possess Lie algebraic structure, and then Poisson's theory on holonomic conservative mechanical systems is used by these systems; and Lagrange-Maxwell mechanico-electrical systems possess Lie admitted algebraic structure, and then Poisson's theory on holonomic conservative mechanical systems is halfway used by these systems. Two examples are discussed to illustrate these results. Key words: algebraic structure, Poisson integral method, mechamico-electrical system PACS: 02.20.Q IntroductionIt is well known that many methods have been developed to seek invariants of mechanical and physical systems. methods, the invariants of the mechanical systems are studied by using algebraic structure seems to have an additional advantage of having a straightforward extension to the corresponding quantum mechanical systems, 15 and algebraic structure and Poisson's theory are carried out to relativistic Birkhoffian systems by Fu et al. [16][17][18] These methods while applied successfully to the mechanico-electrical systems have not been explored so far. In this letter, we make an effort in this direction and demonstrate the applications of the algebraic structure and Poisson's theory of dynamical systems to mechanico-electrical systems. Generalized Hamilton canonical equations of mechanico-electrical systemsMechanico-electrical systems are systems in which a mechanical process and an electromagnetic process are coupled. The mechanical part, constituted by N particles, is described by n generalized coordinates q l (l=1,…,n), and the electrodynamical part, constituted by m electric circuits, is described by m electric quantities e k (k=1, …,m). Consider a mechanico-electrical system composed of m circuits consisting of linear conductors and capacitors. Let i k denote the current, u k denote the electronic potential, R k denote the resistance, and C k denote the capacitance. Then the Lagrangian of the mechanico-electrical 1 Corresponding author: Jing-Li Fu.